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A Non Linear Spectral Graph Neural Network Simulator for More Stable and Accurate Rollouts

Salman N. Salman, Sergey A. Shteingolts, Ron Levie, Dan Mendels

TL;DR

The paper addresses the instability and limited long-range accuracy of traditional graph neural network simulators for molecular dynamics by introducing nonlinear spectral filters (NLSF) that operate in a global eigenmode basis. Through systematic evaluation on four disordered elastic network datasets, the authors demonstrate that NLSF-based simulators outperform spatial and linear spectral counterparts, with particularly strong gains in predicting global properties such as Poisson's ratio and in maintaining accurate rollouts over longer horizons. The key finding is that nonlinear mixing of spectral modes enables efficient capture of slow, collective dynamics, while a norm-based stabilization improves generalization and reduces parameter count. This work suggests a practical path toward more stable, accurate, and data-efficient ML-based simulators suitable for high-throughput screening and inverse design in complex materials and biomolecular systems.

Abstract

Molecular dynamics (MD) simulations are a central tool in science and engineering enabling the study of dynamical behavior and the link between microscopic structure and macroscopic function. Their high computational cost, however, has motivated extensive efforts to develop accelerated alternatives. A promising approach is the use of machine-learning-based simulators that allow for substantially larger time steps than conventional MD. Among these, graph neural network (GNN)-based methods have been found to be especially attractive given that they naturally encode the inductive bias of interacting particle systems, however current architectures remain limited in accuracy and stability. In particular, standard message-passing schemes struggle to efficiently propagate long-range information. Here, we investigate whether spectral-GNN simulators can overcome these limitations by explicitly representing a simulated system in a global eigenmode basis, and therefore better capture long-range and collective behavior. Focusing on disordered elastic networks, model systems for complex materials and biological structures such as proteins, we compare spatial, linear spectral, and nonlinear spectral GNN architectures. We find that while spectral representations alone are not sufficient, nonlinear spectral models substantially outperform alternatives. By learning both the time-dependent dynamics and the mixing of eigenmodes, these models more accurately capture the system's slow, global modes, which dominate macroscopic behavior. This leads to a marked reduction in systematic particle-position error and significantly improved prediction of global physical properties.

A Non Linear Spectral Graph Neural Network Simulator for More Stable and Accurate Rollouts

TL;DR

The paper addresses the instability and limited long-range accuracy of traditional graph neural network simulators for molecular dynamics by introducing nonlinear spectral filters (NLSF) that operate in a global eigenmode basis. Through systematic evaluation on four disordered elastic network datasets, the authors demonstrate that NLSF-based simulators outperform spatial and linear spectral counterparts, with particularly strong gains in predicting global properties such as Poisson's ratio and in maintaining accurate rollouts over longer horizons. The key finding is that nonlinear mixing of spectral modes enables efficient capture of slow, collective dynamics, while a norm-based stabilization improves generalization and reduces parameter count. This work suggests a practical path toward more stable, accurate, and data-efficient ML-based simulators suitable for high-throughput screening and inverse design in complex materials and biomolecular systems.

Abstract

Molecular dynamics (MD) simulations are a central tool in science and engineering enabling the study of dynamical behavior and the link between microscopic structure and macroscopic function. Their high computational cost, however, has motivated extensive efforts to develop accelerated alternatives. A promising approach is the use of machine-learning-based simulators that allow for substantially larger time steps than conventional MD. Among these, graph neural network (GNN)-based methods have been found to be especially attractive given that they naturally encode the inductive bias of interacting particle systems, however current architectures remain limited in accuracy and stability. In particular, standard message-passing schemes struggle to efficiently propagate long-range information. Here, we investigate whether spectral-GNN simulators can overcome these limitations by explicitly representing a simulated system in a global eigenmode basis, and therefore better capture long-range and collective behavior. Focusing on disordered elastic networks, model systems for complex materials and biological structures such as proteins, we compare spatial, linear spectral, and nonlinear spectral GNN architectures. We find that while spectral representations alone are not sufficient, nonlinear spectral models substantially outperform alternatives. By learning both the time-dependent dynamics and the mixing of eigenmodes, these models more accurately capture the system's slow, global modes, which dominate macroscopic behavior. This leads to a marked reduction in systematic particle-position error and significantly improved prediction of global physical properties.
Paper Structure (12 sections, 12 equations, 3 figures, 2 tables)

This paper contains 12 sections, 12 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Datasets
  4. Models
  5. Spatial model
  6. Spectral Filters on Graphs
  7. Non-linear Spectral Filter
  8. Spatio-spectral Models
  9. Computational Details
  10. Results
  11. Discussion and Conclusion
  12. Supplementary Information

Figures (3)

  • Figure 1: The simulator model pipeline with an encoder-processor-decoder architecture. $\varepsilon^x,\varepsilon^e$ are encoders for the node and edge features, respectively. The GNN block takes the various models defined in the Models section. GNN blocks can be stacked. $\delta^x$ is the decoder that maps the output to the accelerations. The update segment constructs the predicted graph from the accelerations. The networks at the top represent samples (left to right) from the datasets "NODE-OPT," "BOND-PRU," and "NOI-PRU, respectively."
  • Figure 2: Performance of the spectral models vs. the spatial model as a function of rollout steps. The first column is the average $L_2$ over several trials between the predicted positions at each rollout step and the ground truth positions at that step. The second column is the average $R^2$ over trials of the predicted and ground truth Poisson's ratio at the given rollout steps. The third column contains the parity plots of Poisson's ratio for a norm-NLSF model (the most accurate model), along with its $R^2$ value as measured at rollout step 100.
  • Figure 3: Upper row. From left to right is Average $R^2$ as a function of history on NODE-OPT, BOND-PRU-OPT, NOI-PRU (from left to right). Lower row. From left to right is Average $L_2$ of the node positions at rollout step 50 as a function of history on NODE-OPT, BOND-PRU-OPT, NOI-PRU. All models were tested on $100$ unseen trajectories. If a model's predictions have $R^2 <0$, we map them all to zero. $L_2 > 1$ are not shown.